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(-2*x + (2*ArcTan[Sqrt[a]*x])/Sqrt[a] + x*Log[1 + a*x^2])/2

Integrate[Log[Sqrt[1 + a*x^2]], x] ==
(-2*x + (2*ArcTan[Sqrt[a]*x])/Sqrt[a] + x*Log[1 + a*x^2])/2 

                          2
Integrate[Log[Sqrt[1 + a x ]], x] == 

       2 ArcTan[Sqrt[a] x]                2
-2 x + ------------------- + x Log[1 + a x ]
             Sqrt[a]
--------------------------------------------
                     2
Time to compute: 0.02 second
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